The nth Prime Page will now find any of the first 2.623&dot;1015 primes or
π(x) for x up to 1017.

Is that truly possible? Can every positive integer have an
associated Prime Curio? No. If we list
everything that any one person (perhaps yourself?) found
interesting about primes, it would be a finite list, because we
are finite (we can do a finite number of things at once and
live for a finite period of time). Yes, we can write an
algorithm for assigning "curious" properties to integers, but
using an algorithm would defeat the intent of our definition of
curios. Since the number of humans is finite, it follows
then that the number of curios must also be finite -- but the
set of integers is infinite!

Our proof that every integer has an associated Prime Curio
is what is called a semantical paradox. Perhaps the
oldest of these is from Epimenides the Cretan who said that:

All Cretans were liars and all other statements made by Cretans were certainly
lies.

Here is a simpler form of this paradox: suppose I say "I am
lying." If I am lying when I say this, then I am telling the
truth; and if I am telling the truth when I say it, then I am
lying. Epimenides probably lived in the fifth or sixth
century BC and is most likely the philosopher referred to by
the Apostle Paul in Titus 1:12. The stories of
Epimenides' life are so fanciful (e.g., that he lived for
hundreds of years and once slept for 57 years) that little is
truly known about him.

Our proof is actually a version of a more recent
semantical paradox sent by G. G. Berry of the
Bodleian Library to Bertrand Russell in a letter dated 21
December 1904. (This letter is reprinted in
[Garciadiego92].) You will often find Berry's paradox stated
as "every integer is interesting." If you reread our proof,
you will be able to reconstruct the "proof" of Berry's paradox.

Berry is also credited with the invention of the greeting
card paradox--he would introduce himself with a card that on
one side said:

The statement on the other side of this card is false.

and on the other said:

The statement on the other side of this card is true.

If you think through these statements, then you will see we
have another version of the Epimenides paradox: if either of
the statements are true, then they must be false as well.

A slightly later version of these paradoxes is Richard's
paradox (1906). His paradox can be stated in the form:

Every positive integer can be uniquely defined using at most
100 keystrokes on a typewriter.

To "prove" this statement you create the type of paradox above
by considering the set S of all integers that cannot be so
described. By the well ordering principal this set has a
least member, and is in fact:

The least positive integer that cannot be described in at
most 100 keystrokes.

But of course we just described it uniquely using less that 100
keystrokes, so to avoid a contradiction the set S must be
empty! Yet, we can also easily disprove this statement because
a typewriter has less that 200 keys (probably closer to 100),
and it follows that 100 keystrokes can describe less than
200100 integers, so Richard's paradox cannot be true.

Russell discussed each of these paradoxes (and several more)
in his "Mathematical logic as based on the theory of types
[Russell1908] (reprinted in [Heijenoort67]) and concludes that
they do not affect the logical calculus which is incapable of
expressing their character.

Again, these are semantical paradoxes unlike Russell's
famous paradox of the set of all sets that do not contain
themselves. This paradox is often recast as a question
about a barber:

If there is a town in which the barber shaves (exactly) those
who do not shave themselves, then does the barber shave
himself?

If he does shave himself, then he does not; and if he does not,
then he does.